Forcing a unique minimum spanning tree and a unique shortest path
For researchers in combinatorial optimization and complexity theory, this paper clarifies the boundary between tractable and intractable forcing problems, showing that some are easier than previously studied cases.
The paper studies the complexity of finding minimum forcing and anti-forcing sets for shortest s-t path and minimum spanning tree problems. It shows these problems are tractable except for the minimum anti-forcing set for shortest s-t paths, which is NP-complete, and provides fixed-parameter tractable algorithms for that case.
A forcing set $S$ in a combinatorial problem is a set of elements such that there is a unique solution that contains all the elements in $S$. An anti-forcing set is the symmetric concept: a set $S$ of elements is called an anti-forcing set if there is a unique solution disjoint from $S$. There are extensive studies on the computational complexity of finding a minimum forcing set in various combinatorial problems, and the known results indicate that many problems are harder than their classical counterparts: the decision version of finding a minimum forcing set for perfect matchings is NP-complete [Adams et al., Discrete Mathematics, 2004], and that of finding a minimum forcing set for satisfying assignments for 3CNF formulas is $Σ_2^P$-complete [Hatami-Maserrat, Discrete Applied Mathematics, 2005]. In this paper, we investigate the complexity of finding minimum forcing and anti-forcing sets for the shortest $s$-$t$ path problem and the minimum-weight spanning tree problem. We show that, unlike the aforementioned results, these problems are tractable, with the exception of the decision version of finding a minimum anti-forcing set for shortest $s$-$t$ paths, which is NP-complete. To complement this intractability result, we design fixed-parameter tractable algorithms for finding a minimum anti-forcing set for shortest $s$-$t$ paths.